Helmholtz Force in Canonical Ensembles

• Helmholtz force is defined as the derivative of the excess Helmholtz free energy with respect to system size in canonical ensembles with fixed order parameters.
• It differs from the Casimir force by being ensemble-dependent and can be either attractive or repulsive based on temperature, magnetization, and boundary conditions.
• Exact studies using Ising, Gaussian, and Nagle–Kardar models demonstrate its sensitivity to finite-size effects and the global constraint imposed on the order parameter.

Helmholtz force (HF) is the fluctuation-induced force associated with the canonical ensemble, in which the total or average order parameter is constrained rather than controlled by an external conjugate field. In the recent statistical-mechanical literature, HF is defined as the derivative with respect to the confining size $L$ of the excess Helmholtz free energy at fixed temperature and fixed total magnetization $M$, or fixed average order parameter $m$. It is the canonical-ensemble counterpart of the Casimir force, which is defined in the grand-canonical ensemble at fixed field $h$. Exact studies in the one-dimensional Ising model, the Nagle–Kardar model, and the three-dimensional Gaussian model show that HF is generally ensemble-dependent, can differ qualitatively from the Casimir force for the same geometry and boundary conditions, and may be either attractive or repulsive depending on temperature, conserved order parameter, and boundary conditions (Dantchev et al., 2023, Dantchev et al., 2024, Dantchev et al., 21 Aug 2025, Dantchev et al., 17 Mar 2026).

1. Canonical definition and thermodynamic status

For a film geometry $\infty^{d-1}\times L$, the canonical definition used in the Ising and Nagle–Kardar studies is

$$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$

with excess Helmholtz free energy

$$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$

In the Gaussian-model formulation the notation is

$$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$

with

$$a_{\rm ex}^{(\tau)}(L,T,m)\equiv a^{(\tau)}(L,T,m)-L a_{\rm bulk}(T,m).$$

The two notational conventions encode the same thermodynamic construction: the force is obtained from the excess Helmholtz free energy, not from the grand-canonical or Gibbs potential (Dantchev et al., 2023, Dantchev et al., 17 Mar 2026).

The controlled variables are the essential distinction. In the grand-canonical ensemble, $T$ and $M$0 are fixed and the order parameter fluctuates. In the canonical ensemble, $M$1 and $M$2 or $M$3 are fixed, so the admissible fluctuations are restricted by the global constraint. This restriction alters finite-size thermodynamics and therefore alters the force. In the sign convention used in the Ising studies, $M$4 is attractive and $M$5 is repulsive (Dantchev et al., 2023).

Physically, HF measures how the free-energy cost of maintaining a fixed total or average order parameter changes as the confinement length changes. The recent exact papers treat this as a genuinely distinct observable rather than a reformulation of the Casimir problem. Their common conclusion is that ensemble choice is itself a control parameter for fluctuation-induced forces in finite systems (Dantchev et al., 2024, Dantchev et al., 17 Mar 2026).

2. Canonical construction and ensemble inequivalence

The canonical partition function is imposed by a global constraint. In the lattice formulations this appears as a Kronecker delta or delta function restricting the total magnetization: $M$6 For the three-dimensional Gaussian model the canonical partition function is written as

$M$7

and, using

$M$8

it becomes a Fourier projection of the grand-canonical partition function evaluated at an imaginary field: $M$9 The one-dimensional Ising papers use the same structural idea, replacing the Kronecker constraint by an integral over $m$0 and projecting the grand-canonical partition function onto fixed $m$1 (Dantchev et al., 17 Mar 2026, Dantchev et al., 2023).

This construction makes explicit why HF and the Casimir force need not coincide. For finite systems the canonical and grand-canonical ensembles correspond to different physical conditions, because one fixes $m$2 and the other fixes $m$3. The one-dimensional Ising analysis further establishes that, in the thermodynamic limit, the bulk Helmholtz and Gibbs free energies are related by Legendre transformation, but the leading finite-size corrections differ strongly: $m$4 whereas in the grand-canonical ensemble

$m$5

The papers therefore identify finite-size inequivalence, rather than bulk thermodynamics alone, as the origin of the difference between HF and the Casimir force (Dantchev et al., 2023).

In the Nagle–Kardar model, which includes long-range equivalent-neighbor interactions, the inequivalence is even sharper. That study states that the canonical and grand-canonical ensembles are not equivalent for finite systems and, due to the long-range interactions, are not equivalent in the thermodynamic limit in the usual sense either. HF and the Casimir force are accordingly treated there as distinct fluctuation-induced forces tied to different ensembles (Dantchev et al., 21 Aug 2025).

3. Exact solution frameworks and scaling structure

The recent exact literature develops HF through three main analytic routes. In the one-dimensional Ising chain, the fixed-$m$6 problem is solved by transfer-matrix methods, Fourier projection onto fixed magnetization, Chebyshev-polynomial representations, and exact reduction to Gauss hypergeometric functions. In the scaling regime, those hypergeometric functions reduce to modified Bessel functions, yielding explicit scaling expressions for the Helmholtz force (Dantchev et al., 2023, Dantchev et al., 2024).

For periodic boundary conditions in the one-dimensional Ising model, one exact canonical partition function is

$m$7

while the Dirichlet/free-boundary study derives an exact fixed-$m$8 partition function in hypergeometric form and then simplifies it using contiguous relations. In both cases the force is extracted from the finite-size Helmholtz free energy and then written in scaling form (Dantchev et al., 2023, Dantchev et al., 2024).

In the three-dimensional Gaussian model, both the excess Helmholtz free energy and the Helmholtz force obey finite-size scaling for all four boundary conditions studied: $m$9

$h$0

The corresponding force scaling function is obtained from the excess free-energy scaling function by

$h$1

For the Gaussian model in $h$2, the paper uses $h$3 and $h$4, so $h$5 (Dantchev et al., 17 Mar 2026).

A recurrent exact feature is that the canonical constraint enters as an additional $h$6-dependent contribution to the free energy. Whether that contribution survives bulk subtraction depends on the boundary condition and the mode structure. This mechanism is explicit in the Gaussian model and underlies the sharp difference between boundary conditions that preserve a uniform mode and those that suppress or distort it (Dantchev et al., 17 Mar 2026).

4. Boundary conditions and sign structure

Exact results show that HF has no universal sign rule analogous to the usual boundary-condition heuristic often invoked for the Casimir force. Its behavior is boundary-condition dependent, ensemble dependent, and frequently $h$7-dependent.

Model and boundary condition	HF behavior	Relation to Casimir force
1D Ising, periodic	Attractive or repulsive depending on $h$8 and $h$9	Casimir always attractive
1D Ising, antiperiodic	Changes sign with $\infty^{d-1}\times L$0 and $\infty^{d-1}\times L$1	Casimir always repulsive
1D Ising, Dirichlet/free	Attractive or repulsive depending on $\infty^{d-1}\times L$2 and $\infty^{d-1}\times L$3	Casimir zero at $\infty^{d-1}\times L$4, attractive for $\infty^{d-1}\times L$5
Gaussian, DD	Attractive or repulsive depending on $\infty^{d-1}\times L$6 and $\infty^{d-1}\times L$7	Casimir always attractive
Gaussian, ND	Always repulsive	Casimir changes sign with $\infty^{d-1}\times L$8
Gaussian, NN	Always attractive, independent of $\infty^{d-1}\times L$9	Coincides with Casimir
Gaussian, periodic	Always attractive, independent of $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$0	Coincides with Casimir

For the one-dimensional Ising model with Dirichlet/free boundary conditions, the low-$$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$1 asymptotics yield an explicit threshold: $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$2 Near $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$3, the HF is attractive for $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$4 and repulsive for $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$5. The same study states that for small $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$6 the Helmholtz force is always repulsive, for moderate $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$7 and suitable $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$8 it can change sign from repulsive to attractive, and for very large $$\beta F_{\rm H}^{(\zeta)}(L,T,M)\equiv- \frac{\partial}{\partial L}f_{\rm ex}^{(\zeta)}(L,T,M),$$9 it tends to zero (Dantchev et al., 2024).

The Gaussian model provides the cleanest exact separation of cases. Under Dirichlet–Dirichlet boundary conditions, the Casimir force is always attractive, while HF can be attractive or repulsive as a function of $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$0 and $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$1. Under Neumann–Dirichlet conditions, HF is always repulsive, whereas the Casimir force changes sign from repulsive to attractive with increase of $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$2. Under periodic and Neumann–Neumann boundary conditions, the two forces coincide; the Casimir force does not depend on $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$3, the Helmholtz force does not depend on $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$4, and both are always attractive (Dantchev et al., 17 Mar 2026).

The structural explanation given in the Gaussian paper is boundary-mode selective. Under periodic and Neumann–Neumann boundary conditions, a uniform mode exists and the constant field or fixed total order parameter couples only to that mode; its contribution is bulk-like and disappears from the excess free energy after bulk subtraction. Under Dirichlet–Dirichlet and Neumann–Dirichlet boundary conditions, the boundaries enforce nonuniform profiles, so the fixed-$$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$5 constraint contributes to the excess free energy and generates a distinct HF (Dantchev et al., 17 Mar 2026).

5. The Nagle–Kardar model and cancellation of long-range contributions

The Nagle–Kardar model supplies a distinct exact result: the complete disappearance of the long-range coupling from HF after the excess Helmholtz subtraction. The model is a one-dimensional Ising chain with nearest-neighbor coupling $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$6, equivalent-neighbor long-range ferromagnetic interaction $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$7, and periodic boundary conditions. In the canonical ensemble the Hamiltonian is constrained by

$$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$8

for $$f_{\rm ex}^{(\zeta)}(L,T,M)\equiv f^{(\zeta)}(L,T,M)-L f_H(T,m), \qquad m=\lim_{L,A\to\infty}\frac{M}{LA}.$$9 and $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$0 (Dantchev et al., 21 Aug 2025).

The central exact statement of that paper is that the $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$1-dependent part of the canonical free energy cancels between the finite contribution and the bulk Helmholtz subtraction. The authors therefore conclude that the Helmholtz force does not depend on $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$2, and that the NK-model HF is exactly the same as the HF of the one-dimensional Ising model with fixed magnetization. This sharply contrasts with the grand-canonical Casimir force of the same model, which is strongly affected by the long-range coupling and by the critical line and tricritical point (Dantchev et al., 21 Aug 2025).

The same work shows that HF changes sign as a function of temperature and magnetization. For negative $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$3, the force is repulsive and increases approximately linearly. For large positive $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$4, the force is again repulsive for the displayed values $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$5. Between those regimes, the force develops a negative minimum, that is, an attractive regime. The depth of that attractive minimum increases as $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$6 decreases; the authors state that the force has a deeper negative minimum for smaller values of $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$7 (Dantchev et al., 21 Aug 2025).

Unlike the Casimir force in the NK model, HF is not presented as carrying a special anomaly tied directly to the critical line or tricritical point. The paper explicitly associates this with the cancellation of the $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$8-dependence. A plausible implication is that, in this model, the canonical fluctuation-induced force is governed primarily by the short-range coupling $$\beta F_{\rm Helmholtz}^{(\tau)}(L,T,m)\equiv- \frac{\partial}{\partial L}a_{\rm ex}^{(\tau)}(L,T,m),$$9 and the conserved magnetization $$a_{\rm ex}^{(\tau)}(L,T,m)\equiv a^{(\tau)}(L,T,m)-L a_{\rm bulk}(T,m).$$0, not by the long-range part that organizes the grand-canonical phase diagram (Dantchev et al., 21 Aug 2025).

6. Conceptual implications, scope, and nomenclature

Across the exact models considered so far, HF is not a minor variant of the Casimir force. It is the force appropriate to a different ensemble and therefore to a different experimental or theoretical control protocol. The exact papers consistently show that fixing $$a_{\rm ex}^{(\tau)}(L,T,m)\equiv a^{(\tau)}(L,T,m)-L a_{\rm bulk}(T,m).$$1 or $$a_{\rm ex}^{(\tau)}(L,T,m)\equiv a^{(\tau)}(L,T,m)-L a_{\rm bulk}(T,m).$$2 can change not only the magnitude but also the sign of the fluctuation-induced force. In the words of the Nagle–Kardar study, the dependence of HF and the Casimir force on tunable variables allows for control of the sign of these forces, as well as their magnitude (Dantchev et al., 21 Aug 2025).

The present exact results are model-specific. The Gaussian paper is explicit that its formulas are exact for the three-dimensional Gaussian model in the scaling regime with the specified slab geometry and boundary conditions, and it does not claim exact universality beyond that model. It does, however, state that the qualitative ensemble dependence is expected more generally, especially when boundaries break translational invariance and force nonuniform order-parameter profiles (Dantchev et al., 17 Mar 2026).

A common source of confusion is nomenclature. “Helmholtz force” in this literature refers to a fluctuation-induced force derived from the Helmholtz free energy in a canonical ensemble. It is not the subject of the paper "Mean value properties of solutions to the Helmholtz and modified Helmholtz equations" (Kuznetsov, 2021). That work studies the PDEs

$$a_{\rm ex}^{(\tau)}(L,T,m)\equiv a^{(\tau)}(L,T,m)-L a_{\rm bulk}(T,m).$$3

and explicitly does not address Helmholtz force directly (Kuznetsov, 2021).

In current arXiv usage within statistical mechanics, HF therefore denotes the canonical-ensemble fluctuation-induced force in confined systems. Its defining features are the derivative with respect to system size of an excess Helmholtz free energy, the presence of a global order-parameter constraint, and a pronounced sensitivity to ensemble choice, boundary conditions, and conserved magnetization (Dantchev et al., 2023, Dantchev et al., 17 Mar 2026).